Fluid Transport · Rotating Equipment

Positive Displacement Pump Sizing: Reciprocating & Rotary Fundamentals

How a fixed volume per stroke or revolution sets the flow, why slip and volumetric efficiency matter, how to size brake horsepower, and why every PD pump needs a discharge relief valve. Governing equipment standards: API 674 / 675 / 676.

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Flow vs pressure

Nearly vertical curve

PD flow is set by geometry & speed, almost independent of discharge pressure.

Volumetric efficiency

90–97% typical

Slip (3–10%) from valve/packing leakage and fluid compressibility.

Safety rule

Relief valve required

A PD pump cannot deadhead — a discharge PSV is mandatory.

Contents

Use this guide when you need to:

  • Size a reciprocating (plunger) or rotary PD pump.
  • Estimate delivered flow from displacement and slip.
  • Compute brake horsepower at a given differential.
  • Specify the mandatory discharge relief valve.

1. How Positive-Displacement Pumps Work

A positive-displacement (PD) pump traps a fixed volume of liquid in a chamber and physically pushes it from suction to discharge. Each stroke (reciprocating) or revolution (rotary) delivers essentially the same volume regardless of the discharge pressure. The pump does not "build head" the way a centrifugal pump does — it simply moves the swept volume, and pressure rises to whatever the downstream system resistance demands.

This gives PD pumps a nearly vertical pump curve: flow is set by displacement geometry and speed, while pressure floats. Three consequences follow directly:

  • High pressure at low flow is easy — a small plunger at modest speed can reach thousands of psi.
  • Flow is self-metering — turn-down is done by changing speed or stroke, not by throttling.
  • It cannot deadhead — close the discharge and pressure climbs until something fails, so a relief valve is mandatory (Section 5).

Where PD pumps fit

Centrifugal pumps own the broad mid-range of higher flow at moderate head. PD pumps take over when the duty is low flow at very high pressure, when precise metered dosing is required, or when the liquid is too viscous for a centrifugal to stay efficient.

Key parameters

Parameter Symbol Units Definition
Plunger diameterdinBore of the plunger/piston (reciprocating)
Stroke lengthLinDistance the plunger sweeps each stroke
Number of plungersn—Simplex 1, duplex 2, triplex 3, quintuplex 5
Displacement per revDin³/revSwept volume of the rotor set per turn (rotary)
SpeedNrpmCrank speed (recip) or shaft speed (rotary)
Displacement flowQdispgpmTheoretical swept volume per unit time
Delivered flowQgpmActual flow after slip
Volumetric efficiencyηv—Q / Qdisp = 1 − slip fraction
Differential pressureΔPpsiDischarge minus suction pressure
Mechanical efficiencyηmech—Hydraulic power / shaft power
Brake horsepowerBHPhpShaft power the driver must supply

2. Reciprocating vs Rotary Positive-Displacement Pumps

PD pumps split into two families by how they trap and move the fixed volume.

Reciprocating pumps (API 674 power, API 675 controlled-volume)

A plunger, piston, or diaphragm reciprocates in a bore. On the suction stroke the inlet check valve opens and the chamber fills; on the discharge stroke the outlet check opens and the trapped volume is expelled. Because each stroke is discrete, the flow pulsates — pulsation dampeners are usually fitted, and multiple plungers (duplex/triplex/quintuplex) smooth the combined delivery.

  • Plunger / piston power pumps (API 674): very high pressure at low flow — high-pressure injection, glycol charge, hydraulic boost.
  • Diaphragm / controlled-volume metering pumps (API 675): precise, repeatable dosing volume per stroke — chemical injection, odorant, polymer dosing.

Rotary pumps (API 676)

Meshing rotors carry liquid in the spaces between them from suction to discharge, delivering a fixed volume per shaft revolution with a much smoother flow than reciprocating. Variants:

  • Gear (internal/external): lube and seal oils, resins.
  • Screw (twin/triple): high-viscosity heavy crude, fuel oil, asphalt.
  • Lobe: gentle handling of shear-sensitive or sanitary fluids.
  • Vane / progressive-cavity: light hydrocarbons and slurries / solids-laden fluids respectively.
TraitReciprocatingRotary
Flow characterPulsating (needs dampeners)Smooth
Best atVery high pressure, low flowViscous liquids, moderate pressure
Displacement set byA · L · n per strokeD per revolution
Slip behaviorValve/packing leakage + compressibilityClearance leakage (rises with ΔP, falls with viscosity)
StandardAPI 674 / API 675API 676

3. Displacement & Brake-Power Relations

The relations below are fundamental positive-displacement definitions — they follow directly from geometry and the standard hydraulic-power identity. They are not proprietary to any one standard; API 674/675/676 are the governing equipment standards for the pumps themselves.

Reciprocating displacement

Plunger area: A = π/4 · d² (in²) Displacement flow: Qdisp = A · L · n · N / 231 (gpm) Where: d = plunger diameter (in) L = stroke length (in) n = number of plungers N = crank speed (rpm) 231 = in³ per US gallon

Rotary displacement

Displacement flow: Qdisp = D · N / 231 (gpm) Where: D = displacement per revolution (in³/rev) N = shaft speed (rpm)

Delivered flow

Reciprocating: Q = Qdisp · ηv Rotary: Q = Qdisp − slip ηv = volumetric efficiency = 1 − slip fraction

Required brake horsepower

BHP = Q · ΔP / (1714 · ηmech) (hp) Where: Q = delivered flow (gpm) ΔP = differential pressure (psi) 1714 = unit constant for gpm · psi → hydraulic hp ηmech = mechanical efficiency (~0.85–0.92)
Why 1714? Hydraulic power = Q·ΔP. With Q in gpm and ΔP in psi, 1 hp = 1714 gpm·psi. Dividing by ηmech converts hydraulic power to the shaft (brake) power the driver must supply.

4. Slip & Volumetric Efficiency

The pump never delivers its full theoretical displacement. The shortfall is slip:

  • Leakage back past suction/discharge valves, plunger packing, or rotor clearances.
  • Fluid compressibility — part of each stroke is spent compressing the trapped liquid rather than expelling it.

Volumetric efficiency ηv = 1 − slip fraction captures this. Two behaviors are universal:

  • Slip rises with ΔP — higher pressure drives more leakage back through clearances, so ηv falls as discharge pressure climbs.
  • Slip falls with viscosity — a thicker liquid seals clearances better, which is exactly why rotary screw pumps shine on heavy crude and oils.
Practice: Reciprocating power pumps typically run 90–97% volumetric efficiency. Rotary pumps are usually characterized by a slip flow read from the manufacturer's curve at the rated differential pressure and viscosity, rather than a single ηv number.

5. The Mandatory Relief Valve & Governing Standards

Because a PD pump moves a fixed volume regardless of resistance, it cannot deadhead. If a downstream valve closes, the pump keeps trying to push its displacement into a closed volume and pressure climbs almost instantly until the casing, piping, seal, packing, or driver fails.

⚠ Safety rule: Every positive-displacement pump must have a discharge relief (pressure-safety) valve, set below the lowest component rating and sized to pass the full pump capacity at the relieving pressure. This is non-negotiable PD practice and is reflected in API 674/675/676.

Governing equipment standards

  • API 674 — Positive Displacement Pumps: Reciprocating (power pumps).
  • API 675 — Positive Displacement Pumps: Controlled-Volume (metering / diaphragm).
  • API 676 — Positive Displacement Pumps: Rotary (gear / screw / lobe).
  • Hydraulic Institute (HI 6.x) — reciprocating and rotary pump nomenclature, test, and application.
Honesty note: The displacement, slip/volumetric-efficiency, and brake-power relations on this page are fundamental positive-displacement theory — universal, derived from geometry and the hydraulic-power identity. The API 674/675/676 PDFs were not on hand when this page was prepared, so no API-specific clause number or proprietary coefficient has been verified against the source documents. Treat the API references as the governing equipment standards for these pumps; consult the current editions for design, materials, testing, and relief-sizing requirements.

6. Worked Example — Triplex Plunger Pump

Size the delivered flow and brake horsepower for a triplex reciprocating power pump. This is the exact case the calculator's self-test reproduces.

Given: Plunger diameter d = 2 in Stroke L = 3 in Plungers n = 3 (triplex) Speed N = 360 rpm Volumetric efficiency ηv = 95% Differential pressure ΔP = 500 psi Mechanical efficiency ηmech = 90% Step 1 — Plunger area: A = π/4 · d² = π/4 · (2)² = 3.1416 in² Step 2 — Displacement flow: Qdisp = A · L · n · N / 231 = 3.1416 · 3 · 3 · 360 / 231 = 44.06 gpm Step 3 — Delivered flow: Q = Qdisp · ηv = 44.06 · 0.95 = 41.86 gpm Step 4 — Brake horsepower: BHP = Q · ΔP / (1714 · ηmech) = 41.86 · 500 / (1714 · 0.90) = 13.57 hp
Result: The pump delivers 41.86 gpm at 500 psi and requires 13.57 brake horsepower. Select a driver with margin above this, and provide a discharge relief valve sized for the full 44 gpm displacement.

Common mistakes to avoid

  • ❌ Omitting the discharge relief valve — a PD pump cannot deadhead.
  • ❌ Confusing displacement flow with delivered flow (forgetting slip).
  • ❌ Sizing the driver on hydraulic power without dividing by ηmech.
  • ❌ Ignoring pulsation (reciprocating) — specify dampeners and check NPSH at peak instantaneous flow.
  • ❌ Assuming ηv is constant — it falls as ΔP rises.
  • ❌ Using 231 incorrectly — it converts in³ to US gallons; keep all lengths in inches.

Key references

  • API 674 – Positive Displacement Pumps — Reciprocating
  • API 675 – Positive Displacement Pumps — Controlled-Volume
  • API 676 – Positive Displacement Pumps — Rotary
  • Hydraulic Institute (HI 6.1–6.5) – Reciprocating & rotary pump standards
  • Displacement & brake-power relations: fundamental PD theory (API PDFs not verified here).

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Frequently Asked Questions

What is the difference between a reciprocating and a rotary positive-displacement pump?

A reciprocating pump (plunger, piston, or diaphragm — API 674/675) displaces a fixed volume each stroke as the plunger sweeps its bore, giving a pulsating flow ideal for very high pressure at low flow. A rotary pump (gear, screw, lobe, vane, or progressive-cavity — API 676) displaces a fixed volume per shaft revolution between meshing rotors, giving a smoother flow well suited to viscous liquids.

What are volumetric efficiency and slip in a positive-displacement pump?

Volumetric efficiency is the delivered flow divided by theoretical displacement; the shortfall is slip — leakage past valves, packing, or rotor clearances plus fluid compressibility. Slip rises with differential pressure and falls with viscosity.

Why does a positive-displacement pump require a discharge relief valve?

A PD pump moves a fixed volume regardless of discharge resistance, so it cannot deadhead. If the discharge is blocked it keeps building pressure until something fails. A discharge relief valve set below the lowest component rating and sized for full capacity is mandatory.